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Question
A straight line passes through the point (α, β) and this point bisects the portion of the line intercepted between the axes. Show that the equation of the straight line is \[\frac{x}{2 \alpha} + \frac{y}{2 \beta} = 1\].
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Solution
The equation of the line with intercepts a and b is \[\frac{x}{a} + \frac{y}{b} = 1\]
This line intersects the axes at A (a, 0) and B (0, b).
Here, (α, β) is the midpoint of AB.
\[\therefore \alpha = \frac{a + 0}{2}, \beta = \frac{0 + b}{2}\]
\[ \Rightarrow a = 2\alpha, b = 2\beta\]
Hence, the equation of the line is \[\frac{x}{2\alpha} + \frac{y}{2\beta} = 1\]
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